Noncommutative blowups of elliptic algebras
type: Item
instance of: scholarly article
title: Noncommutative blowups of elliptic algebras
zbMATH Open document ID: 1331.14004
author: D. Rogalski
author: J. T. Stafford
author: Susan J. Sierra
published in: Algebras and Representation Theory
publication date: 2015-06-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1308.2216
review text: The article studies the noncommutative analogue of the ordinary blowing up in (commutative) algebraic geometry, and it chooses the graded rings approach to noncommutative projective algebraic geometry. The authors show that there is a noncommutative analogue of the blowing-up process, and they give a lot of properties for the resulting algebras. Let \(\mathbb K\) be an algebraically closed ground field and let \(T=\mathbb K\oplus T_1\oplus T_2\oplus\dots\) be a connected graded (cg) domain. \(T\) is called an \textit{elliptic algebra} if there exists a central element \(g\in T_1\) s.t. \(T/gT\) is a \textit{twisted homogeneous coordinate ring} (TCR) \(B(E,\mathcal M,\tau)\) on an elliptic curve \(E\) with \(\mathcal M\) ample and \(\tau\in\text{Aut}(E)\) of infinite order. A TCR is \(B(E,\mathcal M,\tau)=\bigoplus_{n\geq 0}H^0(E,\mathcal L_n)\), \(\mathcal L_n=\mathcal L\oplus\mathcal L^\tau\otimes\cdots\otimes\mathcal L^{\tau^{n-1}}\), and multiplication given by \(f\star g=f\otimes(\tau^m)^\ast(g)\) for \(f\in B_m\) and \(g\in B_n\). The elliptic algebras are the noncommutative analogues of anti-canonical coordinate rings of projective (Fano- or del Pezzo) surfaces. The degree of \(T\) is defined to be \(\deg T=\mu=\deg\mathcal M=\dim T_1-1\). When \(R\) is a noetherian graded domain, its \textit{graded quotient ring} \(Q_{gr}(R)\) is given by inverting all nonzero homogeneous elements. \(R\) is called a \textit{maximal order} of there is no order \(R\subsetneqq R^\prime\subseteq Q_{gr}(R)\) with \(a R^\prime b\subseteq R\) for nonzero \(a,b\in R\). (\(S^{(r)}\) denotes the homogeneous elements of degree \(r\)). The main result in the article is that one can blow up \(T\) in divisors on the elliptic curve \(E\) to obtain new elliptic algebras: Let \(\mathbf d\) be an effective divisor on \(E\) with \(\deg\mathbf d\leq\mu\). Then \(T(\mathbf d)\) is a graded subalgebra of \(T\) with properties given in the main theorem as follows. With \(T,\mu,\mathbf d\) as above,…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 18E15
zbMATH DE Number: 6445654
zbMATH Keywords: noncommutative projective geometry
zbMATH Keywords: noncommutative surfaces
zbMATH Keywords: Sklyanin algebras
zbMATH Keywords: noncommutative blowing up
zbMATH Keywords: Noetherian graded rings
zbMATH Keywords: sporadic ideals
zbMATH Keywords: divisor layering
zbMATH Keywords: graded quotient ring
zbMATH Keywords: twisted homogeneous coordinate ring
zbMATH Keywords: elliptic algebra
zbMATH Keywords: exceptional line modules
zbMATH Keywords: Godie torsion module
MaRDI profile type: Publication
OpenAlex ID: W2155816241
arXiv ID: 1308.2216
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DOI: 10.1007/S10468-014-9506-7
description: scientific article